*Witt theorems for Grassmannians and Lie Incidence Geometry.*

Bruce Cooperstein,
University of California, Santa Cruz.

The concept of singular independence and local independence will be defined for subgraphs of the collinearity graph of a Lie incidence geometry among which are the Grassmannians. I will also introduce the notion of a parabolic subgeometry of a Lie incidence geometry and a parabolic subgraph of the shadow of an apartment. For a Grassmannian G(n,k) a parabolic subgeometry is nothing more than the subgeometry of k-spaces incident with a flag of the underlying PG(n-1).

It will be shown that a subgraph of a Grassmannian geometry or a half spin geometry which is singular independent and isomorphic to a parabolic subgraph is parabolic. It will also be proved that a subgraph of a spin geometry which is locally independent and isomorphic to a parabolic subgraph is parabolic. As a corollary it will be proved that any subgeometry which is isomorphic to a parabolic subgeometry is, in fact, parabolic and all such subspaces will be classified.